Properties

Label 2-384-32.29-c1-0-3
Degree $2$
Conductor $384$
Sign $0.936 - 0.351i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)3-s + (0.184 + 0.444i)5-s + (0.134 − 0.134i)7-s + (0.707 + 0.707i)9-s + (4.83 − 2.00i)11-s + (−0.237 + 0.573i)13-s + 0.480i·15-s + 5.70i·17-s + (0.459 − 1.10i)19-s + (0.175 − 0.0728i)21-s + (4.07 + 4.07i)23-s + (3.37 − 3.37i)25-s + (0.382 + 0.923i)27-s + (−2.33 − 0.966i)29-s − 10.2·31-s + ⋯
L(s)  = 1  + (0.533 + 0.220i)3-s + (0.0823 + 0.198i)5-s + (0.0508 − 0.0508i)7-s + (0.235 + 0.235i)9-s + (1.45 − 0.603i)11-s + (−0.0658 + 0.159i)13-s + 0.124i·15-s + 1.38i·17-s + (0.105 − 0.254i)19-s + (0.0383 − 0.0158i)21-s + (0.850 + 0.850i)23-s + (0.674 − 0.674i)25-s + (0.0736 + 0.177i)27-s + (−0.433 − 0.179i)29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.936 - 0.351i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.936 - 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70773 + 0.310034i\)
\(L(\frac12)\) \(\approx\) \(1.70773 + 0.310034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.923 - 0.382i)T \)
good5 \( 1 + (-0.184 - 0.444i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.134 + 0.134i)T - 7iT^{2} \)
11 \( 1 + (-4.83 + 2.00i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.237 - 0.573i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 5.70iT - 17T^{2} \)
19 \( 1 + (-0.459 + 1.10i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.07 - 4.07i)T + 23iT^{2} \)
29 \( 1 + (2.33 + 0.966i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (3.05 + 7.37i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.877 + 0.877i)T + 41iT^{2} \)
43 \( 1 + (-2.15 + 0.893i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 4.94iT - 47T^{2} \)
53 \( 1 + (-9.74 + 4.03i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.73 + 11.4i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (9.46 + 3.92i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.79 + 1.98i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (4.32 - 4.32i)T - 71iT^{2} \)
73 \( 1 + (6.12 + 6.12i)T + 73iT^{2} \)
79 \( 1 + 11.9iT - 79T^{2} \)
83 \( 1 + (1.59 - 3.86i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.43 - 8.43i)T - 89iT^{2} \)
97 \( 1 + 9.21T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.19729221655615839318946961150, −10.59765892668586256168643514117, −9.249922280726533180464202651758, −8.924423793830576181541277026088, −7.68722300268713735248353331920, −6.68719816352671755460924147435, −5.65626982860170755011043010355, −4.17087040569845628512505813647, −3.33874802905927312881759088766, −1.67015478280331614974741485914, 1.44337242133926301487863037248, 2.98217856182731216419603402603, 4.24387137602891532316532333764, 5.39327884477050989736228801741, 6.85104997781084776124917784840, 7.34106396002709057588087797114, 8.849839145135126936273427361307, 9.158096641517046465056392073530, 10.25150357411831316131676486624, 11.44933237081876988271358002998

Graph of the $Z$-function along the critical line